1. Bayesian Brain: Dynamic Causal Modelling (DCM) This material was modified from Uta Noppeney et al. (Functional Imaging Lab, Wellcome Dept. of Imaging Neuroscience, Institute of Neurology, University College London)

2. Bounded rationality

3. System 1  Fast  Intuitive, associative  heuristics & biases System 2  Slow (lazy)  Deliberate, ‘reasoning’  Rational

4. Bounded rationality limbic system and brainstem (system 1) neocortex (system 2)

5.  Seeing order in randomness  Mental corner cutting  Misinterpretation of incomplete data  Halo effect  False consensus effect  Group think  Self serving bias  Sunk cost fallacy  Cognitive dissonance reduction System 1: very prone to biases  Confirmation bias  Authority bias  Small numbers fallacy  In-group bias  Recall bias  Anchoring bias  Inaccurate covariation detection  Distortions due to plausibility

6.  Bounded Rationality  The Small Numbers Problem of Individual Experience  Prone to See Patterns Even in Random Data  Critical Thinking  Decision Supports  Research Large Ns > individual experience Controls reduce bias The “Human” Problem Evidence-Based Practice

7. “It’s hard to tell the signal from the noise. The story the data tell us is often the one we’d like to hear, and we usually make sure it has a happy ending. It is when we deny our role in the process that the odds of failure rise.” Nate Silver Evidence-based deciison-making

9. 전문가들이 절대 일어나지 않을 것이라고 주장한 사건 가운데 약 15% 가 실제로 일어난다. ( 전문가들이 반드시 일어날 것이라고 한 사건의 약 25% 가 일어나지 않는다.)

10. We don’t like uncertainty!

11. Nate Silver

12. 스카우팅 유망 선수 ( 베이스볼 아메리카 ) vs. 각 선수들의 팀 기여도 계산

13. 538 ( 대통령 선거인단 수 )= 435 ( 하원의원 )+ 100 ( 상원의원 )+ 3 ( 워싱턴 D.C)

18. 당신 ( 아내 ) 이 출장을 마치고 집에 와보니, 처음 보는 속옷이 당신 옷장 서랍 속에 들어있다. 나의 배우자 ( 남편 ) 가 날 속이고 바람을 피우고 있을 확률은 얼마나 될까 ?

19. 사전 확률 : 남편이 바람을 피울 확률의 초기 추정치 (P(A)) 새로운 사건 발생 : 수수께끼의 속옷이 발견됐다 (P(B)). 남편이 바람을 피운다는 조건 아래에서 속옷이 등장했을 확률 (P(B|A)) 사후 확률 당신이 속옷을 발견했다는 조건 아래에서 남편이 바람을 피우고 있을 가능성에 대한 수정된 추정치 (P(A|B))

20. ‘ 표본 외 ’ (Out-of-sample) 라는 함정

25. 움직이는 과녁을 맞혀라 !

28. System analyses in functional neuroimaging System analyses in functional neuroimaging Functional integration Analyses of inter-regional effects: what are the interactions between the elements of a given neuronal system? Functional integration Analyses of inter-regional effects: what are the interactions between the elements of a given neuronal system? Functional connectivity = the temporal correlation between spatially remote neurophysiological events Functional connectivity = the temporal correlation between spatially remote neurophysiological events Effective connectivity = the influence that the elements of a neuronal system exert over another Effective connectivity = the influence that the elements of a neuronal system exert over another Functional specialisation Analyses of regionally specific effects: which areas constitute a neuronal system? Functional specialisation Analyses of regionally specific effects: which areas constitute a neuronal system? MODEL-free MODEL-dependent

29. Approaches to functional integration Functional Connectivity Eigenimage analysis and PCA Nonlinear PCA ICA Effective Connectivity Psychophysiological Interactions MAR and State space Models Structure Equation Models Volterra Models Dynamic Causal Models

30. Psychophysiological interactions stimuli Set Context source target X Modulation of stimulus-specific responses Context-sensitiveconnectivity source source target target

31. V1 V4 BA37 STG BA39 Contextual inputs Stimulus-free - u 2 (t) {e.g. cognitive set/time} Perturbing inputs Stimuli-bound u 1 (t) {e.g. visual words} The aim of Dynamic Causal Modeling (DCM) Functional integration and the modulation of specific pathways yyyyy

32. Hemodynamic model The bilinear model neuronal changes latent connectivity induced response induced connectivity Neuronal model Conceptual overview BOLD y y y Input u(t) activity z 2 (t) Neural activity z 1 (t) activity z 3 (t) c1c1 b 23 a 12 neuronal states

33. Rational statistical inference (Bayes, Laplace) Posterior probability LikelihoodPrior probability Sum over space of hypotheses

34. Constraints on Connections Biophysical parameters Models of Responses in a single region Neuronal interactions Bayesian estimation Conceptual overview posterior  likelihood ∙ prior

35. Example: linear dynamic system LG left LG right RVFLVF FG right FG left LG = lingual gyrus FG = fusiform gyrus Visual input in the - left (LVF) - right (RVF) visual field. z1z1 z2z2 z4z4 z3z3 u2u2 u1u1 state changes effective connectivity external inputs system state input parameters

36. Extension: bilinear dynamic system LG left LG right RVFLVF FG right FG left z1z1 z2z2 z4z4 z3z3 u2u2 u1u1 CONTEXT u3u3

37. Bilinear state equation in DCM state changes intrinsic connectivity m external inputs system state direct inputs modulation of connectivity

38. Hemodynamic model The bilinear model neuronal changes latent connectivity induced response induced connectivity Neuronal model Conceptual overview BOLD y y y Input u(t) activity z 2 (t) activity z 1 (t) activity z 3 (t) c1c1 b 23 a 12 neuronal states

39. important for model fitting, but of no interest for statistical inference The hemodynamic “Balloon” model 5 hemodynamic parameters: Empirically determined prior distributions. Computed separately for each area (like the neural parameters).

40. stimulus function u modelled BOLD response observation model hidden states state equation parameters Combining the neural and hemodynamic states gives the complete forward model. An observation model includes measurement error e and confounds X (e.g. drift). Bayesian parameter estimation by means of a Levenberg-Marquardt gradient ascent, embedded into an EM algorithm. Result: Gaussian a posteriori parameter distributions, characterised by mean η θ|y and covariance C θ|y. Overview: parameter estimation η θ|y neural state equation

41. Constraints on Connections Biophysical parameters Models of Responses in a single region Neuronal interactions Bayesian estimation Overview: parameter estimation posterior  likelihood ∙ prior

42. needed for Bayesian estimation, embody constraints on parameter estimation express our prior knowledge or “belief” about parameters of the model hemodynamic parameters: empirical priors temporal scaling: principled prior coupling parameters: shrinkage priors Priors in DCM posterior  likelihood ∙ prior Bayes Theorem

43. Priors in DCM Shrinkage priors for coupling parameters ( η =0) → conservative estimates! Principled priors: –System stability: in the absence of input, the neuronal states must return to a stable mode –Constraints on prior variance of intrinsic connections (A): Probability <0.001 of obtaining a non-negative Lyapunov exponent (largest real eigenvalue of the intrinsic coupling matrix) –Self-inhibition: Priors on the decay rate constant σ (ησ=1, Cσ=0.105); these allow for neural transients with a half life in the range of 300 ms to 2 seconds Temporal scaling: Identical in all areas by factorising A and B with σ (a single rate constant for all regions) : all connection strengths are relative to the self-connections.

44. Shrinkage Priors Small & variable effect Large & variable effect Small but clear effect Large & clear effect

45. EM and gradient ascent Bayesian parameter estimation by means of expectation maximisation (EM) –E-step: gradient ascent (Fisher scoring & Levenberg-Marquardt regularisation) to compute (i) the conditional mean η θ|y (= expansion point of gradient ascent), (ii) the conditional covariance C θ|y –M-step: Estimation of hyperparameters i for error covariance components Q i : Note: Gaussian assumptions about the posterior (Laplace approximation)

46. Parameter estimation in DCM Combining the neural and hemodynamic states gives the complete forward model: The observation model includes measurement error  and confounds X (e.g. drift): Bayesian parameter estimation under Gaussian assumptions by means of EM and gradient ascent. Result: Gaussian a posteriori parameter distributions with mean η θ|y and covariance C θ|y. η θ|y

47. The DCM cycle Design a study that allows to investigate that system Extraction of time series from SPMs Parameter estimation for all DCMs considered Bayesian model selection of optimal DCM Statistical test on parameters of optimal model Hypothesis about a neural system Definition of DCMs as system models Data acquisition

48. Planning a DCM-compatible study Suitable experimental design: –preferably multi-factorial (e.g. 2 x 2) –e.g. one factor that varies the driving (sensory) input –and one factor that varies the contextual input Hypothesis and model: –define specific a priori hypothesis –Which alternative models? –which parameters are relevant to test this hypothesis? TR: –as short as possible (optimal: < 2 s)